"Once you've arrived at Lamplight Caverns, Mayor MacCready will stop you at the gates of Little Lamplight. You can convince the mayor to let you in with a Speech challenge or with the Child At Heart perk. Failing that, the only way into Little Lamplight is to complete the Rescue from Paradise side quest.

Once in Little Lamplight, there are two ways to get to Vault 87:

1: Passing through Super Mutant infested Murder Pass; if you chose this option, you must first speak with Mayor MacCready or Princess to get the gate to Murder Pass open.

2: Asking Mayor MacCready about alternate paths will reveal a locked back door to Vault 87. You must ask Joseph to restore power to the computer terminal before you can use it. The door and terminal (average difficulty) are located in the Great Chamber."

PS: In future check the answers section more carefully as this question has been asked numerous times before!